Checking the usual places on the Web doesn’t (right now) yield a short answer to this simple question. I’m worried that I will spend just as long trying not to confuse $n$- with $(n+1)$-cells as I did trying to keep straight the geometer's sphere versus the topologist's sphere.
Context seems to matter, as there are clearly different kinds of cells like Schubert cells, Jordan cells, cubical cells, cells in CW complexes, cells ⊂ hyperspheres, ... a knot talk I was watching takes a different attitude to 2-disks and 3-disks than may be permitted elsewhere, etc….
So, for posterity, could someone please correct me if I'm wrong here:
- a 0-cell is like a point
- a 1-cell is isomorphic to a line segment $\sim [0,1]$. Its image may bend depending on the domain of application.
- A graph is made with 0-cells (nodes) and 1-cells (edges). If there were a self-loop in eg a CW-complex then that self-loop could be considered a 1-cell also.
- a 2-cell is isomorphic to a 2-disk $\sim D^2 = S^1 \text{ along with its interior} = \mathrm{closure}\ S^1$, which is like $\sim [0,1]^2$ except that $[0,1]^2$ squashes all of the curvature into the four corners instead of “evenly distributing” it. A 2-cell can also be thought of as isomorphic to the closure of a triangle or the closure of a 2-simplex (the 2-simplex is the convex hull of $(0,0) \cup (0,1) \cup (1,0)$). Usually we should probably think of 2-disks as coming with a requirement of conformality /smoothness unless we're told otherwise.
- a 3-cell is isomorphic to a solid sphere $= \mathrm{closure}\ S^2$, modulo the above
- a 4-cell is isomorphic to a solid 3-sphere $= \mathrm{closure}\ S^3$, modulo the above
- … and so on …
The cells are not to have holes or punctures in them.
(from MathWorld - A Wolfram Web Resource: wolfram.com)
The difference between a cell and a disk is not geometric but rather contextual. "Cell" is meant to indicate many pieces which abut each other in some larger space—and "disk" could be a total space or could be a patch from a total space, but we aren't supposed to think of a foamy sea of disks—if we were, we would use the word "cell".
